Here is a practice called Nurturing Seeds, related to all this and inspired by some of the writings of Thich Nhat Hahn, which we encourage around here.
Buddhist Practice is usually described as flying upon the twin wings of Samatha (calming thoughts and emotions, illuminating and dropping body-mind) and awareness and understanding of vipassana (insight and awareness primarily into the nature and workings of 'self' and mental functions). That is true in Zen practice no less than most other forms of Buddhist practice.
In a nutshell, Vipassana might be described as insights and awareness, based on Buddhist psychology, as to how the mind works and plays it games. It is an understanding of the Skandhas (form, sensation, perception, mental formation, consciousness ... those words always sung in the Heart Sutra), how our thoughts and emotional reactions arise, how we label and divide the world. We should also understand the Buddha's ideas about how suffering arises within us, which is intimately tied to all that.
However, unlike some schools of Buddhism, in Shikantaza we do not pursue any particular practices --during-- Zazen itself in order to cultivate such vipa?yan? insight ... and much insight naturally arises from Zazen as "Zazen does its thing". Perhaps we might say that, just in "just sitting" Shikantaza ... dropping thoughts of this and that, thus quieting the mind's "mind games" ... we develop a natural sensitivity and understanding of the mind's "mind games" (much like one first comes to really appreciate what "urban noise" is when one first drives out of the city to the middle of the desert or some other truly quiet place).
Apart from "on the Zafu" sitting times, however, in the rest of our Buddhist studies and practice, it is good to contemplate and develop such insight, and come to identify the workings of the Skandhas and such within us day to day.
For example, if you feel an angry or jealous thought arising within you during your day, it is very helpful to identify that as a "bit of temporary mind theatre" and "just the self judging and conflicting with another perceived self". That gives us some distance from the passing emotion, and we no longer see the emotion as quite as inevitable and "true" as we might have before.
For example, in the case of anger ... We need to develop a sensitivity to how anger arises within us, the triggers which tend to set it off, the first feeling of it starting to arise and the cycle it follows until vanishing. We need to catch ourself more and develop the ability to say, "I am feeling the emotion of anger now, but it is only the mind created theater which is present in this moment ... it need not be so." We need to see it as a story the self writes for itself, "catch it" and thus not be "sucked in" and fooled as much. (Most people who feel anger do not realize it is just a mind created bit of theater which can be replaced by something else ... it is not the way things "have to be". E.g., most people think, when they become upset, that they have "reason to be upset, and it is true and justified", not an optional response to the circumstances). That realization and understanding of how our inner theater works is a step to developing the ability to "rewrite and change the story" at will.
So, yes, "samatha/vipassana" are both important.